Editing Covariant transformation (section)

==Dual properties==

Entities that transform covariantly (like basis vectors) and the ones that transform contravariantly (like components of a vector and differential forms) are "almost the same" and yet they are different. They have "dual" properties.
What is behind this, is mathematically known as the [[dual space]] that always goes together with a given linear [[vector space]].

Take any vector space T. A function ''f'' on T is called linear if, for any vectors '''v''', '''w''' and scalar α:
:<math>\begin{align}
  f(\mathbf{v} + \mathbf{w}) &= f(\mathbf{v}) + f(\mathbf{w}) \\
        f(\alpha \mathbf{v}) &= \alpha f(\mathbf{v})
\end{align}</math>

A simple example is the function which assigns a vector the value of one of its components (called a ''projection function''). It has a vector as argument and assigns a real number, the value of a component.

All such ''scalar-valued'' linear functions together form a vector space, called the '''dual space''' of T. The sum ''f+g'' is again a linear function for linear ''f''  and ''g'', and the same holds for scalar multiplication α''f''.

Given a basis <math>\mathbf{e}_i</math> for T, we can define a basis, called the '''dual basis''' for the dual space in a natural way by taking the set of linear functions mentioned above: the projection functions.  Each projection function (indexed by ω) produces the number 1 when applied to one of the basis vectors <math>\mathbf{e}_i</math>.  For example, <math>\omega^0</math> gives a 1 on <math>\mathbf{e}_0</math> and zero elsewhere.  Applying this linear function <math>{\omega}^0</math> to a vector <math>\mathbf{v} =v^i \mathbf{e}_i</math>, gives (using its linearity)
:<math>
  \omega^0(\mathbf{v}) = \omega^0(v^i \mathbf{e}_i) = 
    v^i \omega^0(\mathbf{e}_i) = v^0
</math>
so just the value of the first coordinate. For this reason it is called the '''projection function'''.

There are as many dual basis vectors <math>\omega^i</math> as there are basis vectors <math>\mathbf{e}_i</math>, so the dual space has the same dimension as the linear space itself. It is "almost the same space", except that the elements of the dual space (called '''dual vectors''') transform covariantly and the elements of the tangent vector space transform contravariantly.

Sometimes an extra notation is introduced where the real value of a linear function σ on a tangent vector '''u''' is given as
:<math>\sigma [\mathbf{u}] :=  \langle \sigma, \mathbf{u}\rangle</math>
where <math>\langle\sigma, \mathbf{u}\rangle</math> is a real number. This notation emphasizes the bilinear character of the form. It is linear in σ since that is a linear function and it is linear in '''u''' since that is an element of a vector space.